The context of this problem is the estimation of the distribution of a parameter $v$ given sets of data $A$ and $B$, where $A$ and $B$ are not independent.
Suppose I know $P(v | A)$ and $P(v | B)$. Both of these distributions are definitely not normal (the answer should not make any assumptions about the functional form here)
Suppose I also know that $A$ and $B$ are not independent. In particular, the data come in pairs $(a_i, b_i)$. The exact relationship between $a_i$ and $b_i$ is not known but one can suppose that they are linearly correlated i.e. $a_i \sim N(f + gb_i, \sigma_a^2)$, or alternatively $b_i \sim N(j + ka_i, \sigma_b^2)$ . However, it should not be assumed that the data sets $A$ and $B$ are normally distributed (either jointly or independently).
The question: how do I work out $P(v | A, B)$?
The obvious approach is to try : $P(v | A, B) \propto P(A | v, B) P (v | B)$, however I can't see how to calculate the first term with only the knowledge above.
Another approach is to try $P(v | A, B) \propto P(A, B | v) P(v) = P(A | B, v) P(B | v)$ which doesn't help either.
The inter-relationship of $A$ and $B$ suggests that $P(A | B, f, g) = \prod_i P(a_i | b_i, f, g, \sigma$). The $f$, $g$ and $\sigma$ can be integrated out if needed (with appropriate priors) to get $P(A | B)$. But I can't see how to introduce the relationship between $A$ and $\nu$ into this to get to $P(A | \nu, B)$.
Any help greatly appreciated.
One approach I have considered is
$P(\nu | A, B) = \int P(\nu, X | A, B) \delta(X - \nu)\,dX \stackrel{?}{=} \int P(\nu | A)P(X | B) \delta(X - \nu)\,dX$
$ = P(\nu|A)P(\nu|B)$ but I'm almost certain that the second step isn't valid.